Download Using Neural Gas Networks in Traffic Navigation

Using Neural Gas Networks in Traffic Navigation
Ján Vaščák
Centre for Intelligent Technologies, Technical University in Košice,
Letná 9, 042 00 Košice, Slovakia,
e-mail: [email protected]
Abstract:
The quality of navigation methods for mobile means depends first of all on
description accuracy of their movement in a given area. This paper deals
with Neural Gas (NG) networks whose role is creating of topologies for
complex objects as for instance road networks of municipal
communications where it is necessary to determine relations among
individual elements (in our case a network of mutually interconnected
communication nodes), too. The proposed approach combines besides NG
networks also tree search algorithms, namely A*, hereby enabling to
consider also various restraints, e.g. traffic rules, too. The performance of
the proposed algorithm is shown on the road network of the city Košice in
Slovakia.
Keywords: navigation, Neural Gas networks, tree search algorithms, algorithm A*
1. Introduction
At present there are three basic groups of navigation methods for mobile means:
heuristic, grid and exact algorithms [1]. A typical representative of the first group there
are Bug algorithms. They are simple but suitable only for avoiding only a smaller
number of obstacles, mainly in tasks for preventing immediate collisions. Potential
fields [15, 18] are the most known approaches of the grid algorithms. However, they
require not only creating a potential field of the whole area in advance but in the case of
some changes (e.g. a new obstacle) it will be necessary again newly to create the whole
field and computational efforts are considerably high. The exact algorithms as visibility
graphs or Voronoi diagrams enable to find the best (in our case the shortest) trajectory.
If there is no solution they will be even able to terminate the computation and so to
prevent timeless sub-cycling. Opposite to the potential fields in the case of changes it
will be necessary to do modifications only in the given area. However, their drawback is
that they require very accurate data about obstacles that means a serious problem in
practical applications.
The NG networks are basically graphs, which enable modelling the form of given
patterns, similarly as in Kohonen networks (Self-organizing Maps) but opposite to them
NG networks do not have any definite topology of connections in the output layer. This
fact seems to be very advantageous mainly in the case of non-homogenous areas. Their
ability of accurate description for a given pattern was compared to other kinds of neural
networks and confirmed e.g. in [6, 12, 21]. Therefore it seems to be very suitable to
203
Vol. 2. No. 2. 2009
Acta Technica Jaurinensis
utilize them just for the description of road networks with various types and forms of
communications. Besides, the use of some modified search algorithms enables us to
incorporate into the description various restraints, e.g. traffic rules or traffic density, too.
In addition, convenient graph structure of NG networks enables very efficient searching.
This paper deals with description of NG networks principles, followed by description of
the traffic networks problem and modification of the A* algorithm. In concluding parts
some experiments will be analysed, which were done on a real road network of the city
Košice in Slovakia.
2. Principles of NG Networks
The NG networks are basically derived from Kohonen networks where for the position
change determination of output neurons also the neighbourhood principle is used.
However, opposite to them the neighbourhood is changed in each adaptation step
according to given input [9]. This enhanced adaptation measure enables a relatively free
movement of neurons in the area, which should be described just by their suitable
deployment and it represents analogy to gas spreading in a closed space. Learning of
NG networks mutually combines two learning paradigms – own NG learning and
competitive Hebbian learning.
Let us suppose that we have a predetermined number of points N with help of them we
can describe a given space. They will be denoted as reference vectors w whose elements
are in general space coordinates – in our case they will be two-dimensional. Reference
vectors w divide the area (space) into N parts and in each of them they represent centres
of these parts. Such an approach is known as vector quantization that represents certain
form of coding with help of which we can describe given area and which is the own task
of NG networks. The success of this task is dependent just on suitable deployment of
reference vectors. Therefore, similarly as in Kohonen networks, in individual cycles we
will select points ξ belonging to this area and with their help we will adapt positions of
reference vectors, which define positions of neurons ci of the network output layer A,
i.e. A={c1, c2, …, cN}.
The object of the own learning is adaptation of reference vectors, i.e. calculation of their
change ∆wi(t) in the time t. It is a kind of the multiple-winners learning where the most
change is at the winner vector ws1 but in accordance with the neighbourhood range
h(t,k) also other k neighbours are changed although in a lower measure. The vector ws1
is assigned to such a point, which is the nearest to the input ξ, i.e. arg(min(ξ – wS1)). The
ambition is to reach bigger changes in a broader range at the beginning of the learning
process and continuing with time this trend would decrease till the adaptation end time
tmax. From this reason parameters λp and λz are defined – starting and final where λp λz.
The parameter λ for the given time t is defined as:
λ (t ) = λ p (λz / λ p )t / t
max
(1)
and the neighbourhood range h(t, k) for the kth neighbour is computed as:
⎛ 1− k ⎞
h(t , k ) = exp ⎜
⎟.
⎝ λ (t ) ⎠
204
(2)
Acta Technica Jaurinensis
Vol. 2. No. 2. 2009
The adaptation process is also influenced by the learning parameter γ whose value is
time-dependent according to γp and γz similarly as for λ and the calculation is like in (1).
The entire adaptation process in individual adaptation steps till tmax is following:
1. Sequencing all reference vectors by their distance to ξ into series:
ξ − wS 1 ≤ ξ − wS 2 ≤ … ≤ ξ − wSN .
(3)
2. Adapting all reference vectors by:
∆wSi (t ) = y (t ) ⋅ h(t , k ) ⋅ (ξ − wSi ).
(4)
3. Setting up the time t = t+1 and until t < tmax return to the step 1 else finish the
adaptation.
It is possible to prove that this kind of adaptation principally corresponds to the
optimization of a cost function according to the gradient descent method [10], which
does not exist in Kohonen networks. In other words, the adaptation in NG networks is
usually quicker.
The competitive Hebbian learning [9] serves for constructing a topological structure
among neurons of the output layer. This kind of learning comes from the basic idea of
Hebbian learning, i.e. that the connections whose neurons are activated at the same time
(synchronously) are strengthened. Their change corresponds to the product of the
activation values of these neurons. However, at the same time a competition element is
embedded, too. This means in one adaptation step only one connection is created,
namely between the two closest network points to the input ξ, i.e. between the reference
vectors ws1 and ws2 of the points S1 and S2 (neurons c1 and c2). The distance among
points is usually calculated by Euclidian norm. In [11] it was shown that the topology
created in such a manner corresponds to Delaunay triangulation. Consequently, after
transforming to Voronoi regions, these regions ensure finding an optimal path.
The simplest way for learning NG networks is using a two-stage process where at first
suitable deployment of a given number of points is created and then they are mutually
interconnected using competitive Hebbian learning. However, this approach is possible
only in the case of predefined tmax, which is a considerably limiting circumstance. The
other possibility is based on parallel processing of both learning kinds. However, there
is a danger that Delaunay triangulation will be damaged due to continued changes of
reference vectors. From this reason it is necessary to incorporate a mechanism of
removing obsolete connections, too. For this purpose a process of aging connections is
used where an age v(c1, c2) is assigned to each connection between the neurons c1 and
c2. At the moment of creating a new connection its age is set up to zero as well as in the
case if the algorithm tries again to create this connection (the so-called connection
rejuvenating). The age will be incremented by 1 in all connections among all direct
neighbours of c1 if it becomes again the winner. If the age of a connection reaches the
value T(t) it will be removed. In such a way a risk of invalid connections will be
minimized. As at the adaptation start bigger changes are done it is necessary for T(t) to
be changed in time from smaller to bigger values. It is determined in a similar way as in
(1) where Tp Tz.
205
Vol. 2. No. 2. 2009
Acta Technica Jaurinensis
A learning process by adaptation steps for obtaining a description of a ring (grey
surface) [4] is depicted in Fig. 1. It is possible to observe the influence of parameters
(λp= 10, λz = 0.01, γp = 0.5, γz = 0.005, tmax = 40000, Tp = 20, Tz = 200) with advancing
time, too. From intelligibility reasons in the depictions b – e the connections among
neurons are missing.
In the first steps of learning a large neighborhood of the input ξ contains adapted points,
which is characterised by a massive deployment of a large number of points in a form
similar to the given space. Later, the influence of adaptation parameters h(t, k) and γ(t)
will be weakened and thereby the adaptation will be performed only in close
surroundings of the input ξ, i.e. on a local level individual reference vectors try to cover
the given space uniformly. Such a phenomenon is typical for various forms of described
space [3].
Figure 1. Learning process of a NG network with competitive Hebbian learning on a
ring indicating the number of steps
2.1. Growing NG Networks
A considerable limitation of the previous approach is based on a fact that it is necessary
to determine a fix number of output neurons in advance whose estimation can be
difficult (if at all possible). Besides, further required property could be embedding a
quality criterion directly as an ending condition for learning. These requirements led to
a design of modified NG networks with incremental learning named as Growing NG
(GNG) networks [2] using already mentioned algorithms.
A GNG network starts learning with two starting neurons. For growing their number as
well as their deployment it uses heuristics and the so-called quantization error Ec of
individual neurons ci where squared errors of the vector ws relating to the input ξ will be
accumulated, i.e.:
Ec = ∑ ξ − wS
206
2
(5)
Acta Technica Jaurinensis
Vol. 2. No. 2. 2009
if this neuron is a winner, i.e. a neuron with the biggest error (further point S1). The
learning task is to perform the space quantization in such a way to obtain the minimum
total quantization error E, i.e. ∑Ec for i = 1, …, N. From experimental experience
(heuristics) it can be mostly supposed that inserting further neurons will reduce the error
E and the best result will be obtained if the new point r is ‘close’ to S1. For ‘closeness’
determination there is another heuristic based on using point S2, which is a direct
neighbour having the biggest Ec comparing to another ones. A new reference vector wr
will be then placed between S1 and S2:
wr = ( wS1 + wS 2 ) / 2.
(6)
Consequently, the connection between S1 and S2 will be removed and new two ones
will be created between S1 and r as well as S2 and r. Simultaneously, the errors of
points S1 and S2 will be reduced by values α.Ec1 and α.Ec2 where α is the quantization
error reduction parameter. As it is clear the point r will not fully reduce the total
quantization error analogically to (6) a certain amount of starting error will be assigned
to r, i.e. Er = (Ec1 + Ec2)/2.
Unlike NG networks only the reference vectors of S1 and its direct neighbours are
adapted. We can consider only local adaptation with uniform growing of output neurons
as it is visible in Fig. 2 and it seems to be also a certain advantage from the point of
view of possible simpler learning process analysis. As the adaptation of reference
vectors is an iterative process inserting new points will be done only in adaptation steps
being a natural product of the parameter τ. Further differences to NG networks are
timely constant learning parameters for S1 and its direct neighbours Si, i.e. γS1 and γSi as
well as the maximum age of a connection T. It is supposed in each adaptation step
certain improvement will occur and therefore the quantization error Eci will be always
reduced by the value β.Eci.
The whole learning process of GNG networks is as follows:
1. During initialization two starting neurons are selected arbitrarily. The connection set
is empty.
2. An input signal ξ enters the system and the point S1 with the biggest error Ec1 from
all ci ∈ A will be obtained. Consequently, the point S2 as a direct neighbour of S1
will be determined. The points S1 and S2 will be connected and the connection age
will be set up to zero. If the connection already exists then its age will be set up
again to zero.
For c1 the equation (5) will be used and reference vectors for S1 as well as its direct
neighbours will be adapted by:
∆wSi = γ i ⋅ (ξ − wSi )
(7)
for i = 1, 2, … and the age of their connections will be incremented.
3. If a connection reaches the age T it will be removed as well as all points without any
connections.
207
Vol. 2. No. 2. 2009
Acta Technica Jaurinensis
4. If the adaptation step reaches a natural product of τ (if no then next step 6) a new
point r (6) will be inserted and connections and errors for S1, S2 and r will be
modified.
5. For each neuron ci the quantization error Eci is reduced by the value β.Eci.
6. If the ending condition is not yet fulfilled (e.g. maximum network dimension or
minimum error) then continue next adaptation step and go to the step 2.
In Fig. 2 there is depicted learning process of a GNG network by individual adaptation
steps again on the same example of a ring (see Fig. 1) [4] (τ = 300, γS1 = 0.05, γSi =
0.0006, α = 0.5, β = 0.0005, T = 88, N = 100). Comparing Fig. 1 and Fig. 2 it is
possible to see differences of learning NG and GNG networks. However, the obtained
results are almost identical (see Fig. h).
Figure 2. Learning process of a GNG network on a ring indicating the number of steps
3. Utilization and Modification of NG Networks for Purposes of Path
Planning
Still nowadays data for traffic navigation systems are created by a complicated way
with considerable portion of manual work either in preparing maps or directly on place
in measuring orientation points using GPS. In our case it is possible to almost fully
automate this preparation stage (besides inserting data about one-way roads and other
entry restrictions). Using a colour filter only communications are extracted from the
map (Fig. 3) and this kind of information will be a direct basis (training set) for
learning. For our purposes we used a GNG network and experiments were done on the
map of city Košice.
In Fig. 3 there is a part of a primarily learned network (blue colour) together with the
real state of communications (black colour). We can see that this network partially:
•
208
connects communications without any real connection, which is wrong,
Acta Technica Jaurinensis
•
Vol. 2. No. 2. 2009
is redundant, i.e. it contains too many points, which can be omitted and thereby
we can get a simpler network structure.
Figure 3. A part of a primarily learned GNG network
During removing wrong connections it is necessary to distinguish between really
incorrect connections as e.g. the connections a, b in Fig. 4 and principally acceptable
connections c, d. Acceptable connections represent a principal existence of a road only
they are not able accurately to describe its form. From this reason additional points will
be inserted better to form it. For separating these two cases a new modification of the
Bug2 algorithm [8] was proposed [16] where a search oval is created with the radius ρ
covering the investigated connection (see Fig. 4). If in this oval a continuous connection
exists between the end points of the investigated connection then the connection is
acceptable else it will be removed.
Further step is removing redundant connections. In Fig. 3 it can be seen straightforward
roads are described not by only one connection but by several shorter connections, too.
All intermediate connections are dispensable because they can be substituted by a
longer one, which causes lower memory efforts and shortening the path search. Usually,
using reduction mechanisms a considerably simpler network is obtained.
Since, in a traffic network there are both bidirectional and one-way roads the network
from Fig. 3 will be doubled and connections will be given orientations. Consequently,
for one-way communications (including rotaries) the prohibited directions will be
removed. Only this stage requires manual activity. Finally, the connections will be
given the information about their length, which is in other words the path cost. The last
two kinds of information, i.e. orientation and path cost are fundamental for path search
algorithms.
209
Vol. 2. No. 2. 2009
Acta Technica Jaurinensis
Figure 4. Identification of wrong connections
Creating a description of the road network in the form of a tree structure using NG
networks represents only the first stage in the navigation task. Next stage is preformed
by tree search algorithms utilizing the structure of a NG network as a convenient data
source where they solve the task to find the best path between two points. For this
reason the A* algorithm was used being able to find the shortest path with a minimum
number of browsing and in the case of its absence also being able to give a message [8].
However, it is necessary still to take some modifications of the A* algorithm [19] to
keep traffic rules, namely:
•
prohibition of turning in a node,
•
the so-called P-problem.
Since in the bidirectional communications there is each connection doubled with reverse
orientations this would enable turning in next node (point), which means turning in a
road. Therefore, being used the current connection its reverse orientation is temporarily
cancelled.
As seen from Fig. 5 the so-called P-problem resembles to the character P. There is a
problem of inability to find a path although it exists indeed. Let us suppose a car is on
the connection between the points 1 and 2 and tries again to come back to the point 1. In
such a case the algorithm will stop although there is a solution in the form
2→3→4→5→2→1 (principles of A* as well as its modifications are more detailed
described in [20]). To prevent this problem the coding of the whole NG network was
changed in such a way the path will not be searched by the points but by their
connections [19]. In other words, neurons of the NG network will not represent
communication points (crossings and curves) but connections between these points. In
our case for Fig. 5 the solution will look like 23→34→45→52→21.
210
Acta Technica Jaurinensis
Vol. 2. No. 2. 2009
Figure 5. Description of the P-problem
4. Setting up Learning Parameters and Experiments
There are in total eight parameters whose setting up influences not only the learning
process but also the entire quality of the resulting space description. In following we
will summarize them and introduce some remarks concluded from experiments. Namely
there are these parameters for GNG networks:
•
number of adaptation steps given by the time tmax,
•
maximum number of neurons (points) of the output layer N,
•
interval of inserting new points τ,
•
maximum age of a connection T,
•
learning parameters γS1 and γSi,
•
parameters for reduction of the quantization error α and β.
By [7] the value for γS1 is chosen considerably smaller than 0,3 – in our case γS1 = 0,05
and for γSi the value will be approximately one tenth, i.e. γSi = 0,006. Further, for our
experiments values of remaining parameters were: T = 100, α = 0,5, β = 0,0005.
The experiments were mainly focused on observing parameters tmax, N and τ because
just these parameters influence most the quality of the created network and interact
mutually. Combining their various values and comparing obtained results following
outcomes can be confirmed:
1. The optimum number of output neurons is approximately one hundredth of the
number of training points.
2. The number of adaptation steps should be from two up to three times bigger as the
number of training points.
The interval of inserting new points should enable inserting all points during the first
2/3 of the adaptation (having enough time for deploying new points to a correct
position), i.e.:
τ≈
2tmax
.
3N
(8)
211
Vol. 2. No. 2. 2009
Acta Technica Jaurinensis
During the experiments a roughly uniform deployment of training points was supposed.
The bigger number of adaptation steps the more accurate the network but also the bigger
computational efforts. Therefore, the proposed values for parameters express a
compromise between descriptive precision and computational speed. Although the
greatest influence on the position of a given neuron is caused by the initial step of
inserting (6) but it will be influenced also by continuous adaptation of its position even
if relating to the magnitudes of γS1 and γSi in a considerably smaller measure. Therefore,
it is necessary to enable a multiple adaptation of each neuron and from this reason tmax
needs to have high values as well as an inserted neuron needs to have possibility of
moderate position correction at least during the last third of the entire adaptation time
tmax.
For needs of creating a GNG network describing the communication network of the city
Košice a training set with 242 470 points was used and other parameters owned these
values: tmax = 900 000, N = 2 000, τ = 300, T = 100, α = 0,5, β = 0,0005, γS1 = 0,05, γSi
= 0,006. The experiments showed the network was able to learn with the same quality
on various types of road networks regardless the form and density as seen in the Fig. 6.
5. Conclusions
The proposed combination of NG networks and exact graph algorithms offers very
advantageous properties for navigation either as an auxiliary for drivers [17] or as a
direct means for navigation of mobile robots with the ability incrementally to modify
space description. The absence of any definite topology of connections in the output
layer (in comparison to Kohonen networks) enables NG networks to model whatever
area of arbitrary complexity without any limitations regarding various restrains, e.g.
forms of roads and traffic rules in the case of communications. This approach enables
including further mechanisms like automatic removing of connections in the case of
one-way roads or merging several independently created networks describing
neighbourhood areas. It is possible to create an overview network with a less detailed
description (like maps with different scales) for purposes of approximate navigation
[13], too. Since the concept of NG networks is general for spaces with arbitrary
dimensions it is also possible to incorporate for instance height data.
The advantage of this approach is based mainly on its two-stage processing. In the first
stage a descriptive network is created although computationally demanding but
necessary only ones, which will be later modified only occasionally using mentioned
mechanisms. Anyway, also in this stage most of activities are automated (opposite to
conventional approaches). In the second stage, which will be used many times, highly
efficient algorithms are used for finding optimum paths whose time efforts do not
exceed 3 seconds in the case of Košice.
The descriptive form of NG networks for contour manifold and heterogeneous spaces
offers further utilization possibilities. It would be probably very perspective to use such
a numerical knowledge representation form for knowledge extraction into rules using
fuzzy logic (because of its nonlinear matter [14]) and its learning (adaptation)
approaches [5] to obtain symbolic form of knowledge, which is necessary for more
complex control and decision tasks.
212
Acta Technica Jaurinensis
Vol. 2. No. 2. 2009
Figure 6. Description of the road network for urban part Košice – North
Acknowledgement
This publication is the result of the project implementation Centre of Information and
Communication Technologies for Knowledge Systems (project number: 26220120020)
supported by the Research & Development Operational Programme funded by the
ERDF.
213
Vol. 2. No. 2. 2009
Acta Technica Jaurinensis
References
[1] Dudek, G., Jenkin M.: Computational Principles of Mobile Robotics, Cambridge
University Press, Cambridge, (2000).
[2] Fritzke, B.: A growing neural gas network learns topologies, In: Advances in
Neural Information Processing Systems 7; MIT Press Cambridge, USA, (1995),
pp. 625-632.
[3] Fritzke, B.: Some competitive learning methods, (1997), pp. 45,
<citeseer.ist.psu.edu/fritzke97some.html> [cit. 22.7. 2008].
[4] Fritzke, B.: Vektorbasierte Neuronale Netze (Prof. thesis in German), Shaker
Verlag, (1998), pp. 157, <http://www.ki.inf.tu-dresden.de/~fritzke/>
[cit. 22.7. 2008].
[5] Heinke, D., Hamker F.H.: Comparing Neural Networks: A Benchmark on Growing
Neural Gas, Growing Cell Structures, and Fuzzy ARTMAP, IEEE Transactions on
Neural Networks, Vol. 9, No. 6, (1998), pp. 1279-1291. [6]
[6] Holmström, J.: Growing Neural Gas (PhD. thesis), Uppsala University Department of Information Technology, (2002), pp. 42. [7]
[7] Johanyák, Z. C., Kovács, Sz.: A brief survey and comparison on various
interpolation based fuzzy reasoning methods, Acta Polytechnica Hungarica, Vol. 3,
No. 1, (2006), pp. 91-105. [5]
[8] Lavalle, S. M.: Planning algorithms, Cambridge University, (2006), pp. 842,
http://planning.cs.uiuc.edu/ [cit. 22.7. 2008].
[9] Martinetz, T. M., Schulte, K. J.: A neural gas network learns topologies, In:
Artificial Neural Networks; North Holland Amsterdam, (1991), pp. 397-402.
[10] Martinetz, T. M.: Competitive Hebbian learning rule forms perfectly topology
preserving maps, In: ICANN - International Conference on Artificial Neural
Networks, Amsterdam, Springer, (1993), pp. 427-434. [11]
[11] Martinetz, T. M.: Selbstorganisierende neuronale Netzwerkmodelle zur
Bewegungssteuerung (in German), Infix Verlag, (1992). [10]
[12] Milano, M., Koumoutsakos, P., Schmidhuber, J.: Self-Organizing Nets for
Optimization, IEEE Transactions on Neural Networks, Vol. 15, No. 3, (2004), pp.
758-765.
[13] Mls. K.: Implicit knowledge in Concept Maps and their Revealing by Spatial
Analysis of Hand-Drawn Maps, Proc. of the Second International Conference on
Concept Mapping – Concept Maps: Theory, Methodology, Technology, Vol. 2,
San José, Costa Rica, (2006).
[14] Oblak, S., Škrjanc, I., Blažič, S.: If approximating nonlinear areas, then consider
fuzzy systems, IEEE Potentials, Vol. 25, No. 6, (2006), pp. 18-23.
[15] Pozna, C., Troester, F., Precup, R.-E., Tar, J. K., Preitl, St.: On the Design of an
Obstacle Avoiding Trajectory: Method and Simulation, Mathematics and
Computers in Simulation, Elsevier Science, Vol. 79, No. 7, (2009), pp. 2211-2226.
[16] Rutrich, M.: Využitie sietí typu Neural Gas v navigácii (MSc. thesis in Slovak), TU
v Košiciach, (2007), pp. 67.
[17] Spalek, J., Dobrucký, B., Lusková, M., Pirník, R.: Modeling of Traffic Flows of
Suburban Agglomerations: Technical and Social Aspects, Applications, The 2nd
International Conference on Knowledge Generation, Communication and
Management: KGCM 2008 ORLANDO, (2008).
214
Acta Technica Jaurinensis
Vol. 2. No. 2. 2009
[18] Vaščák, J., Rutrich, M.: Path Planning in Dynamic Environment Using Fuzzy
Cognitive Maps, In: SAMI – 6th International Symposium on Applied Machine
Intelligence and Informatics, Herľany, Slovakia, January 21-22 (2008), pp. 5-9.
[19]
[19] Vaščák, J., Szászi, T.: Navigácia mobilných robotov pomocou harmonických
potenciálových polí (1) and (2) (in Slovak), In: AT&P Journal, No. 2 and 3, (2007)
pp. 58-60 and 74-75. [18]
[20] Vaščák, J.: Fuzzy cognitive maps in path planning, In: Acta Technica Jaurinensis:
Series intelligentia computatorica, Vol. 1, No. 3, (2008), pp. 467-479.
[21] Yeh, M. F., Chang, K. Ch.: A Self-Organizing CMAC Network With Gray Credit
Assignment, IEEE Transactions on Systems, Man, and Cybernetics – Part B:
Cybernetics, Vol. 36, No. 3, (2006), pp. 623-635.
215